Vector measures on orthocomplemented lattices

Abstract

A relatively orthocomplemented lattice L is a lattice in which every interval is an orthocomplemented sublattice. An orthogonally scattered measure ξ on L is a Hilbert space valued abstract measure over L such that ξ(e) ⊥ ξ(f) whenever e ⊥ fin L. The properties of so generalized c.a.o.s. measures are studied, the representation theorem has been proved: every H-valued c.a.o.s. measure ξ on L is of the form ξ(e) = Φ(e)x, where x ε H, and Φ is a lattice orthohomomorphism from L into Proj (H). The results generalize those in [21]. Their suitability for many applications has been demonstrated, including duality theory for some inductive-projective limits of Hilbert spaces and quantum probability.